Kerr black holes with synchronised hair: an analytic model and dynamical formation

TL;DR

This paper introduces an analytic model for Kerr black holes with synchronised Proca hair, validated against numerical simulations, and explores their formation via superradiance, enhancing understanding of hairy black hole states.

Contribution

It presents a universal analytic model for Kerr black holes with synchronised hair, applicable in the weak hair regime, and compares it with numerical solutions to confirm accuracy.

Findings

The model accurately describes hairy black holes with quasi-Kerr horizons.

Numerical simulations confirm the model's validity in the superradiant domain.

The study identifies conditions for the dynamical emergence of hairy black holes.

Abstract

East and Pretorius (arXiv:1704.04791) have successfully evolved, using fully non-linear numerical simulations, the superradiant instability of the Kerr black hole (BH) triggered by a massive, complex vector field. Evolutions terminate in stationary states of a vector field condensate synchronised with a rotating BH horizon. We show these end points are fundamental states of Kerr BHs with synchronised Proca hair. Motivated by the "experimental data" from these simulations we suggest a universal (i.e. field-spin independent), analytic model for the subset of BHs with sychronised hair that possess a quasi-Kerr horizon, applicable in the weak hair regime. Comparing this model with fully non-linear numerical solutions of BHs with synchronised scalar or Proca hair, we show the model is accurate for hairy BHs that may emerge dynamically from superradiance, whose domain we identify.

Equations

$$M_{(\psi )}=M-M_H,J_{(\psi )}=J-J_H.$$

$$M_H=\frac{1}{2}{T}_H{A}_H+2{\Omega }_H{J}_H.$$

$$j\equiv \frac{J}{M^2},a_H\equiv \frac{A_H}{16\pi M^2},w_H\equiv {\Omega }_{H}M,t_H\equiv 8\pi M{T}_H,$$

$$p\equiv \frac{M_{(\psi )}}{M},q\equiv \frac{J_{(\psi )}}{J}.$$

$$j=\frac{4w_H}{1+4w_H^2},a_H=\frac{1}{1+4w_H^2},t_H=1-4w_H^2,$$

$${\Omega }_H=\frac{M_H}{2J_H}\left(1-\chi \right),$$

$$A_H=8\pi M_H^2\left(1+\chi \right),T_H=\frac{\chi }{4\pi M_H\left(1+\chi \right)},$$

$$d{M}_{(\psi )}={\Omega }_{H}d{J}_{(\psi )}.$$

$$M_{(\psi )}={\Omega }_H{J}_{(\psi )},i.e.M-M_H={\Omega }_H(J-J_H).$$

$$p=w_{H}jq.$$

$$q=p\frac{1+4(1-p{)}^2{w}_H^2}{p+4(1-p{)}^2{w}_H^2},j=\frac{p+4(1-p{)}^2{w}_H^2}{w_H(1+4(1-p{)}^2{w}_H^2)},$$

$$a_H=\frac{(1-p{)}^2}{1+4(1-p{)}^2{w}_H^2},t_H=\frac{1-4(1-p{)}^2{w}_H^2}{1-p}.$$

$$a_H=a_H^{(Kerr)}+\left[\frac{(1+\sqrt{1-j^2}-\frac{1}{2}{j}^2)(1+\sqrt{1-j^2})}{j^2\sqrt{1-j^2}}\right]p^2.$$

$$d{s}^2=-{\sigma }^2(r)N(r)d{t}^2+\frac{d{r}^2}{N(r)}+r^{2}d{\Omega }_2,$$

$$\mathcal{A}=e^{-iwt}\left[f(r)dt+ig(r)dr\right].$$

$$\frac{d}{dr}\left\{r^2\sigma (r)N(r)g(r)\right\}=-\frac{w{r}^{2}f(r)}{\sigma (r)N(r)}.$$

$$d{s}^2=-$$

$$\mathcal{A}=\left(iVdt+\frac{H_{1}}{r}dr+H_{2}d\theta+iH_{3}\sin\theta d\varphi\right)e^{i(m\varphi-wt)}\,,\ \ \ \ \$$

Full text

Introduction. A 50 year-old lingering question in black hole (BH) physics has been the endpoint of the Kerr BH superradiant instability Press and Teukolsky (1972), triggered by massive, bosonic fields Damour et al. (1976); Zouros and Eardley (1979); Detweiler (1980); Brito et al. (2015a). Apart from its theoretical interest, this instability offers an unexpected opportunity for testing the existence of ultra-light bosonic particles suggested by beyond the standard model scenarios, $e.g.$ Arvanitaki et al. (2010). BHs effectively become particle detectors Arvanitaki and Dubovsky (2011), creating a remarkable synergy between strong gravity, particle physics and astrophysics, testable by ongoing/future gravitational waves and electromagnetic observations (see $e.g$ Arvanitaki and Dubovsky (2011); Pani et al. (2012); Witek et al. (2013); Yoshino and Kodama (2012, 2014); Pani and Loeb (2013); Arvanitaki et al. (2015); Brito et al. (2015b); Cunha et al. (2015); Yoshino and Kodama (2015); Vincent et al. (2016); Arvanitaki et al. (2017); Ni et al. (2016); Baryakhtar et al. (2017); Zhou et al. (2017); Franchini et al. (2016); Fujita and Cardoso (2017)).

In a recent breakthrough, East and Pretorius East and Pretorius (2017) reported long term numerical evolutions of this instability, using a Proca field to trigger it. Their evolutions lead to equilibrium states wherein the BH horizon angular velocity, $\Omega_{H}$, synchronises with the phase angular velocity of the (complex) Proca field. This suggests these endpoints belong to the family of Kerr BHs with synchronised Proca hair (KBHsPH), previously constructed as fully non-linear, stationary solutions of the Einstein-Proca system Herdeiro et al. (2016).

In the present letter we establish the equilibrium states obtained dynamically in East and Pretorius (2017) correspond to fundamental states of KBHsPH, making these BHs the first dynamical counter example to the no-hair conjecture Ruffini and Wheeler (1971), in general relativity, with a simple, physically reasonable matter content. Moreover, we identify which of these hairy BHs can be endpoints of superradiance.

More generically, for a subset of BHs with synchronised bosonic hair, including all solutions that may dynamically form from the superradiant instability, we propose an analytic model based on the hypothesis that the horizon is *quasi-*Kerr. This holds when the BH hair is weak, but also even for considerably hairy BHs when the matter field is dilute. Within this model, explicit, universal ($i.e$ field-spin independent) formulas for physical quantities are presented, in terms of a “hairiness” parameter. These provide an analytic handle for further studies of these solutions in this interesting regime.

The quasi-Kerr horizon (qKH) model. Consider a stationary, axisymmetric, asymptotically flat, BH with synchronised hair of a $generic$ bosonic matter field $\psi$, with field mass $\mu$. Synchronisation means that $\Omega_{H}=w/m$ where $\psi\sim e^{-i(wt-m\phi)}\dots$, $\partial/\partial t$, $\partial/\partial\phi$ are the Killing vector fields associated to stationarity and axi-symmetry, respectively, $m\in\mathbb{Z}^{+}$ and $w$ is the field’s frequency. Known examples have been constructed for both scalar Herdeiro and Radu (2014, 2015); Kleihaus et al. (2015); Herdeiro et al. (2015a) and vector matter Herdeiro et al. (2016). The ADM mass and angular momentum are $M$ and $J$, while the corresponding horizon data are $M_{H}$, $J_{H}$ (computed as Komar integrals) together with the area, $A_{H}$, temperature, $T_{H}$ and $\Omega_{H}$. The corresponding matter field data obey (see $e.g$ Herdeiro et al. (2016))

[TABLE]

The following Smarr relation also holds $M=\frac{1}{2}T_{H}A_{H}+2\Omega_{H}[J-J_{(\psi)}]+M_{(\psi)},$ or, equivalently

[TABLE]

Furthermore, the solutions satisfy the first law of thermodynamics $dM=\frac{1}{2}T_{H}dA_{H}+\Omega_{H}dJ.$

It is convenient to define the aforementioned physical quantities normalised by the ADM mass,

[TABLE]

and to introduce two measures of the ’hairiness’, which provide, respectively, the fraction of energy (angular momentum) in the matter field

[TABLE]

In the absence of hair ($p=0=q$), $i.e$ for Kerr,

[TABLE]

with $0\leqslant w_{H}\leqslant{1}/{2}$, the limits corresponding the Schwarzschild solution and extremal Kerr, respectively.

In the presence of hair ($p\neq 0\neq q$) we assume the BH horizon is *quasi-*Kerr. We expect this to hold in the weak hair regime, $p,q\ll 1$ 111When $p,q\rightarrow 0$, the hair becomes a stationary cloud on the Kerr background. For the scalar case these clouds were first discussed in Hod (2012) – see also Hod (2013); Herdeiro and Radu (2014); Hod (2015, 2017).. Under this assumption, we introduce the quasi-Kerr horizon (qKH) model approximating the horizon quantities by those of a Kerr BH, but with the replacements $(M,J)\rightarrow(M_{H},J_{H})$:

[TABLE]

where $\chi\equiv\sqrt{1-{J_{H}^{2}}/{M_{H}^{4}}}$. For the Kerr family, these formulas are exact, but for hairy BHs they are just an approximation.

For hairy BHs (2) still holds; from (1) together with the first law of thermodynamics and the qKH model, we find that the matter field mass and angular momentum obey

[TABLE]

We formally integrate (8) treating $\Omega_{H}$ as an input parameter. This is justified regarding the hairy BH as a composed system of a horizon plus a matter distribution. Then $\Omega_{H}$ characterises the horizon subsystem which is external (albeit coupled) to the “hair” subsystem. Then

[TABLE]

An equivalent form of (9) reads

[TABLE]

Relations (6), (10) give two constraints for the four variables $(p,q,w_{H},j)$. We may choose $p,w_{H}$ as the independent (control) parameters yielding the simple expressions

[TABLE]

which are, thus, predictions of the qKH model that shall be compared against the fully non-linear (numerical) solutions in the next section. Similarly,

[TABLE]

Observe the Kerr relations (5) are recovered as $p=0$.

qKH model $vs.$ numerical solutions. To test the qKH model we compare it with fully non-linear (numerical) BHs with synchronised hair, which are solutions of the corresponding Einstein-matter system. To test universality, we shall consider both the scalar solutions constructed in Herdeiro and Radu (2014, 2015) and Proca solutions. The latter belong to the family introduced in Herdeiro et al. (2016) and are described in the Appendix. The corresponding part of the domain of existence to be explored is shown in Fig. 1 222The numerical results herein are for $m=1$ being obtained from several thousands of solution points for each type of hairy BHs, with typical error estimates $<10^{-3}$..

Fig. 1 shows the neighbourhood of the existence line wherein the hairy BHs reduce to vacuum Kerr ($p=0$). We shall be particularly interested in the hairy BHs that may arise from the growth of the superradiant instability of Kerr. Then, thermodynamics imposes an upper bound of $p\leqslant 1-1/\sqrt{2}\simeq 0.29$. The corresponding BHs exist in the allowed strip between the $p=0$ and $p=0.3$ lines in Fig. 1. The complete domain of existence can be found in Herdeiro and Radu (2014) (Fig. 5 below) for the scalar (Proca) case.

In Fig. 2 we test the qKH model by comparing the model’s results for $\mathcal{Y}=\{j,a_{H},t_{H}\}$ with the data from the numerical solutions. It shows the relative errors $|1-\mathcal{Y}^{(th)}/\mathcal{Y}^{(num)}|$ for the allowed strip with $p\in[0,0.3]$, $\omega_{H}\in[0,0.5]$. This analysis shows that for the solutions with $j<1$, whose importance will be detailed below, the relative errors are below the percent level for all physical quantities, and typically lower for the scalar case. The errors are still only at a few percent level when $p\sim 0.3$ and for low values of $\omega_{H}$, the relative errors are below 1%, even for $p\sim 0.3$. We interpret this unexpected accuracy in the description as a consequence of a separation of scales. Indeed, $\omega_{H}<\mu M$, and $M$ (the gravitational radius of the BH), and $\mu$ (the inverse Compton wavelength of the massive bosonic field) are the two fundamental scales of the problem. When the dimensionless coupling $M\mu$ is small – often called the Newtonian regime – the bosonic field is diluted, even though it may correspond to a large fraction of the total spacetime energy 333Such property, already observed in Brito et al. (2015b), is not found, as far as we know, in other models of hairy BH solutions.. In this regime it is expectable that the BH horizon is not strongly affected by the (mostly far away) bosonic field distribution, being *quasi-*Kerr. This type of separation of scales is often used to alleviate the complexity of physical problems. Two examples in the context of strong gravity appear in the study of spinning BH binaries Gerosa et al. (2015) and the blackfold approach to higher dimensional BHs Emparan et al. (2009).

The analysis of Fig. 2 validates the qKH model for the region where $j<1$. But we emphasise that even for $j>1$ the model can be accurate, especially for small $\omega_{H}$. This is illustrated in Fig. 3 where we exhibit a $a_{H}$ $vs.$ $j$ plot. Even for $j$ approaching 2 (thus for non-Kerr BHs) the analytic model fits well the numerical points, along lines of constant $\omega_{H}$, for both the scalar and Proca case, supporting its (matter field spin) universality.

Dynamical formation from superradiance. The qKH model was suggested by a phenomenological observation in East and Pretorius (2017). In Fig. 3 therein, a Kerr-like approximation was observed to accurately fit the equilibrium state obtained from the numerical simulations of the growth of the superradiant instability, triggered by a Proca field. To establish that such equilibrium states are the hairy BHs with a quasi-Kerr horizon that we have described before, Fig. 4 below mimics Fig. 3 in East and Pretorius (2017), wherein the normalised irreducible mass, $M_{H}$ and $J_{H}$ are plotted. In the formalism introduced above, these quantities translate as $M_{\rm ir}/M=\sqrt{a_{H}}$, $M_{H}/M=1-p$ and $J_{H}/M^{2}=j(1-q)$.

Fig. 4 shows that the qKH model fits accurately our solutions of both Kerr BHs with scalar or Proca hair, as it did fit the equilibrium BHs obtained through numerical evolutions by East and Pretorius in East and Pretorius (2017). This establishes that the equilibrium states obtained through the growth of the superradiant instability are the fundamental states of Proca BHs with synchronised hair, $cf.$ the Appendix, and in particular the ones with a *quasi-*Kerr horizon. It also predicts a similar result for the scalar case: superradiance forms BHs with synchronised scalar hair and a *quasi-*Kerr horizon (see Brito et al. (2015b) for related observations).

We can now investigate how the dynamics of superradiance migrates a vacuum Kerr BH into a BH with synchronised hair. This is exhibited in Fig. 5, for the examples of the numerical evolutions in East and Pretorius (2017). Under the assumption used therein that a single superradiant mode is present (the fastest growing mode), which implies that axisymmetric is a good approximation during the evolution, radiation is negligible East and Pretorius (2017) and the total mass and angular momentum are preserved. The process is thus conservative. This implies that the migration in Fig. 5 – an ADM mass $vs.$ horizon angular velocity plot – occurs along a horizontal line, ending at a hairy BH with the same $j$ as the initial Kerr solution. We have verified there is a unique such solution – $cf.$ the inset in Fig. 5. In particular this implies the equilibrium BH with synchronised hair has $j\leqslant 1$. Solutions with $j\leqslant 1$ exist in a sub-strip of the allowed strip $p<0.3$. This explains our emphasis in this region in Fig. 2. Using eq. (11) one shows, moreover, that $j\leqslant 1\Rightarrow p\lesssim 0.0973$, a stronger bound on the hairiness that can form dynamically from superradiance.

Hairy BHs are entropically favoured. The (vacuum Kerr) $\rightarrow$ (hairy BH) migration is conservative (in the above description), but it is irreversible. Thermodynamics determines the arrow of time. To understand this, we resort to the qKH model, which allows us to show that the hairy BH is always entropically favoured (as observed in Herdeiro and Radu (2014) from the numerical data). Indeed, from the $a_{H}(j,p)$-cubic equation: $a_{H}^{3}-2(1-p)a_{H}^{2}+\left[{j^{2}}/{4}+(1-p)^{2}\right]a_{H}={j^{2}(1-p)^{2}}/{4},$ the solution for small $p$ [up to $\mathcal{O}(p^{3})$] reads

[TABLE]

Thus for the same $M,J$, the hairy BH is entropically favoured over the Kerr BH. In order words, the direction of the migration in Fig. 5 is determined by the second law of thermodynamics.

Remarks. It was observed in Herdeiro and Radu (2014) that there is a non-uniqueness for Kerr BHs with synchronised scalar hair (as there is in the Proca case). Near the Kerr limit, this degeneracy is discrete and of degree two: fixing $M,J$ there exists a hairy BH and a vacuum Kerr BH. The numerical simulations reported in East and Pretorius (2017), together with the qKH analytic model we have proposed and the fundamental states of KBHsPH described in the Appendix, support the conclusion that these degenerate states correspond to the initial (Kerr) and final (hairy) states of the superradiant instability.

When more than one mode (and with different values of $m$) becomes important during the superradiant evolution, the axi-symmetry assumption in East and Pretorius (2017) may not be accurate and less smooth evolutions may occur, $cf.$ the results reported in non-asymptotically flat setups Sanchis-Gual et al. (2016a); Bosch et al. (2016); Sanchis-Gual et al. (2016b). In this case, BHs with synchronised hair and a quasi-Kerr horizon may be transient equilibrium states. One may imagine, for instance, that the equilibrium states dynamically attained in East and Pretorius (2017), with $\omega_{\rm eq}/m=\Omega_{H}^{\rm eq}$, may be driven to evolve by a lower frequency modes, $\omega_{\rm new}/m<\Omega_{H}^{\rm eq}$. An evolution of $\Omega_{H}^{\rm eq}$ towards synchronising it with $\omega_{\rm new}$ is, however, hampered by the existing $\omega_{\rm eq}$ mode. Understanding this process is a relevant open issue 444This process is dissipative and likely more dissipative in the Proca case East (2017)..

Finally, it would be interesting to adapt the qKH model for higher dimensional BHs with synchronised hair Brihaye et al. (2014); Herdeiro et al. (2015b), including in $AdS$ Dias et al. (2011). Another possible application of this model is in a study of the thermodynamics of BHs with synchronised hair.

Acknowledgements. We are very grateful to W. East and F. Pretorius for correspondence and V. Cardoso, P. Cunha and P. Pani for comments. C. H. and E. R. acknowledge funding from the FCT-IF programme. This work was partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013. Computations were performed at the Blafis cluster, in Aveiro University.